a family of wavelets and a new orthogonal …musaelab.ca/pdfs/c2.pdf · a generalisation of the...

A FAMILY OF WAVELETS AND A NEW ORTHOGONAL MULTIRESOLUTIONANALYSIS BASED ON THE NYQUIST CRITERION

H.M. de Oliveira1, Member IEEE, L.R. Soares2, T.H. Falk1, Student M IEEE

CODEC - Communications Research Group1

Departamento de Eletrônica e Sistemas - CTG- UFPELaboratório Digital de Sistemas de Potência2 LDSP- UFPE

C.P. 7800, 50711-970, Recife - PE, Brazile-mail: hmo@npd.ufpe,br, lusoares@npd.ufpe.br, tiagofalk@go.com

ABSTRACT- A generalisation of the Shannon complexwavelet is introduced, which is related to raised cosinefilters. This approach is used to derive a new family oforthogonal complex wavelets based on the Nyquistcriterion for Intersymbolic Interference (ISI)elimination. An orthogonal Multiresolution Analysis(MRA) is presented, showing that the roll-off parametershould be kept below 1/3. The pass-band behaviour ofthe Wavelet Fourier spectrum is examined. The left andright roll-off regions are asymmetric; nevertheless theQ-constant analysis philosophy is maintained. Finally,a generalisation of the (square root) raised cosinewavelets is proposed.Key-words- Multiresolution Analysis, Wavelets,Nyquist Criterion, Intersymbolic Interference (ISI).

1. INTRODUCTION

Wavelet analysis has matured rapidly over the pastyears and has been proved to be valuable for bothscientists and engineers [1,2]. Wavelet transforms haverecently gained numerous applications throughout anamazing number of areas [3, 4]. Another stronglyrelated tool is the Multiresolution analysis (MRA).Since its introduction in 1989 [5], MRA representationhas emerged as a very attractive approach in signalprocessing, providing a local emphasis of features ofimportance to a signal [2, 6, 7]. The purpose of thispaper is twofold: first, to introduce a new family ofwavelets and then to provide a new and completeorthogonal multiresolution analysis. We adopt thesymbol := to denote equals by definition. As usual,Sinc(t):=sin(πt)/πt and Sa(t):=sin(t)/t. The gate functionof length T is denoted by ∏

Tt . Wavelets are denoted

by ψ(t) and scaling functions by φ(t). The paper isorganised as follows. Section 2 generalises the SincMRA. A new orthogonal MRA based on the raisedcosine is introduced in section 3. A new family oforthogonal wavelet is also given. Furthergeneralisations are carried out in section 4. Finally,Section 5 presents the conclusions.

2. A GENERALISED SHANNON WAVELET(RAISED-COSINE WAVELET)

The scaling function for the Shannon MRA (or SincMRA) is given by the sample function:

)()()( tSinctSha =φ . A naive and interestinggeneralisation of the complex Shannon wavelet can bedone by using spectral properties of the raised-cosine

filter [8]. The most used filter in DigitalCommunication Systems, the raised cosine spectrumP(w) with a roll-off factor α, was conceived toeliminate the Intersymbolic Interference (ISI). Itstransfer function is given by

( )

+≥+

2

unidimensional grid of integers, n=±1,±2,±3,… .Thisscaling function φ defines a non-orthogonal MRA.Figure 2 shows the scaling function corresponding to ageneralised Shannon MRA for a few values of α.

Figure 2. Scaling function for the raised cosine wavelet(generalised Shannon scaling function).

3. MULTIRESOLUTION ANALYSIS BASED ONNYQUIST FILTERS

A very simple way to build an orthogonal MRA via theraised cosine spectrum [8] can be accomplished byinvoking Meyer's central condition [6]:

∑∈

=+ΦZn

nwπ

π2

1|)2(| 2 . (4)

Comparing eqn(2) to eqn(4), we choose

)()( wPw =Φ (i.e. a square root of the raised cosinespectrum). Let then

( ).)1(||

)1(||)1(

)1(||0

0

)1(||4

1cos

2

12

1

)(

παπαπα

παπα

απ

π

+≥+

3

∏

∏∏

−−+−

+

−++

−=Φ

B

wwt

B

wwt

B

ww

2)cos(

2)cos(

22)(2

00

00

πθ

πθ

ππ (8)

with parameters πα=:B , α41

:0 =t and

απα

θ4

)1(:0

−−= .

It follows from Definition 1 that( )

−−−+

−−

+−=Φ

παπα

παα

παπα

παα

ππ

,,4

)1(,

41

;,,4

)1(,

41

;

22,0,0,0;)(2

wPCOSwPCOS

BwPCOSw

and therefore( )

−−−+

−−

+−=

παπα

παα

παπα

παα

πφπ

,,4

)1(,

4

1;,,

4

)1(,

4

1;

22,0,0,0;)(2 )(

tpcostpcos

BtpcostdeO

After a somewhat tedious algebraic manipulation, wederive

{ }.t)(.tt)()t(

..

]t)[(Sinc)..()t()deO(

απααπαπ

απ

ααπ

φ

−++−

+−−=

1sin41cos41

14

2

1

112

1

2

(9)A sketch of the above MRA scaling function is shownin figure 4, assuming a few roll-off values.

Figure 4. "de Oliveira" Scaling Function for anOrthogonal MRA (α=0.1, 0.2 and 1/3).

The scaling function )()( tdeOφ can be expressed in amore elegant and compact representation with the helpof the following special functions:

Definition 2. (Special functions); ν is a real number,)(:)( tSinctH ννν = , 0≤ν≤1, and

[ ]{ }t.t)(t

)(t

||:)t( 22112

21

21 sin2cos21

211

2πνννπν

νννν

πνν −+−−

−=Μ

oIt follows that

)()(2 1)( tHtSha =φπ ,

)()()(2 111)( ttHtdeO αααφπ

+−− Μ+= .

Clearly, )()(lim)()(

0

tt ShadeO φφα

=→

.

The low-pass H(.) filter of the MRA can be found byusing the so-called two-scale relationship for thescaling function [7]:

Φ

=Φ

2.

22

1)(

wwHw . (10)

How should H be chosen to make eqn(10) hold?Initially, let us sketch the spectrum of Φ(w) and Φ(w/2)as shown in figure 5.The main idea is to not allow overlapping between theroll-off portions of these spectra. Imposing that2π(1−α)>(1+α)π, it follows that α

4

Defining a shaping pulse

( )

Φ−Φ=

2.2

2

1:)()(

wwwS deO π

π, (14)

the wavelet specified by eqn(12) can be rewritten as( ).)( )(2/)( wSew deOjwdeO −=Ψ The term e-jw/2 accounts

for the wave, while the term S(w) accounts for let.

Figure 6. Sketch of the scaling function spectrum:

(a) scaled version

Φ

2

w (b) translated version ( )π2−Φ w

(c) simultaneous plot of (a) and (b).

From the figure 6, it follows by inspection that:

+≥

+

5

Aiming to investigate the wavelet behaviour, wepropose to separate the Real and Imaginary parts ofs(deO)(t), introducing new functions rpc(.) and ipc(.)

{ }

−ℑ=ℑ

−ℜ=ℜ )

2

1()( )

2

1()( )()()()( tsmtmtsete deOdeOdeOdeO ψψ

(18)Proposition 2. Let Bww +=∆ + 0

)1( : and

Bww −=∆ − 0)1( : ; 0000

)1( : θθ ++=∆ + twBt and

0000)1( : θθ −−=∆ − twBt be auxiliary parameters. Then

( )20

2

}1,1{

)()(

}1,1{

)()(0 sencos)(.cossen.

2

1

tt

twittwt

trpc iii

i

ii

−

∆∆+∆∆−=

∑∑+−∈+−∈

θθ

π

( )20

2

}1,1{

)()(

}1,1{

)()(0 coscos)(.sensen.

2

1

tt

twittwt

tipc iii

i

ii

−

∆∆−+∆∆−=

∑∑+−∈+−∈

θθ

π oProof. Follows from trigonometry identities.At this point, an alternative notation

( ))1()1()1()1( ,,,;)( −+−+ ∆∆∆∆= θθwwtrpctrpc and( ))1()1()1()1( ,,,;)( −+−+ ∆∆∆∆= θθwwtipctipc can be

introduced to explicit the dependence on these newparameters. Handling apart the real and imaginary partsof )()( ts deO , we arrive at

( )

( )

−+++−

+

−+=ℜ

0,2

),1(2),1(2;r0,0),1(),1(2;

2,0),1(),1(;)(2 )(

παπαπαπαπ

παπαππ

tpctrpc

trpctse deO

(19)Applying now proposition 2, after many algebraicmanipulations:

( ) =ℜ )(2 )( tse deOπ { })()()()(

2

1 )1(2)1(2

11)1()1(2 tttt

αα

αααα

+−

−++− Μ+Μ+Η−Η= ,

(20)and ( ) ( ))2/1()( )()( −ℜ=ℜ tsete deOdeOψ .The analysis of the imaginary part can be done in asimilar way.Definition 3. (special functions); ν is a real number,

tt

tHνπ

νπνν

)cos(:)( = , 0≤ν≤1, and

[ ]{ }tttsin

tt 22112

21

21 cos.)(2)(21

||21:)(1

2πνννπν

νννν

πνν −−−−

−=Μ

oThe imaginary part of the wavelet can be found by

( )=ℑ )(2 )( tsm deOπ { })()()()(

2

1 )1(2)1(2

11)1()1(2 tttt

αα

αααα

+−

−++− Μ+Μ+Η−Η=

(21)and ( ) ( ))2/1()( )()( −ℑ=ℑ tsmtm deOdeOψ .

The real part (as well as the imaginary part) of thecomplex wavelet )()( tdeOψ are plotted in figure 8, forα=0.1, 0.2 and 1/3.

( ))()( te deOψℜ

(a)( ))()( tm deOψℑ

(b)Figure 8. Wavelet )()( tdeOψ :

(a) real part and (b) imaginary part.

4. FURTHER GENERALISATIONS

Generally, the approach presented in the last section isnot restricted to raised cosine filters.

Algorithm of MRA Construction. Let P(w) be a realband-limited function, P(w)=0 w>2π, which satisfiesthe vestigial side band symmetry condition, i.e.,

{ } ππ

π |

6

square of both members and adding equations for eachinteger n,

ααπαλπ dnwPnwZnZn

∫ ∑∑∈∈

+=+Φ1

0

2 );2()(|)2(| .

Since P(w;α) is a Nyquist filter,

π

απ2

1);2( =+∑

∈Zn

nwP and the proof follows. o

The most interesting case of such generalisationcorresponds to a "weighting" of square-root-raise-cosine filter.Corollary. (Generalised raised-cosine MRA). IfP(w;α) is the raised cosine spectrum with a(continuous) roll-off parameter α, i.e.,

( )

+≥+